Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions

@article{Tongas2007AffineLA,
  title={Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions},
  author={A. Tongas and Dimitrios Tsoubelis and Pavlos Xenitidis},
  journal={Journal of Physics A},
  year={2007},
  volume={40},
  pages={13353-13384}
}
We consider lattice equations on which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and a conservation law. A systematic analysis of the Lie point and the generalized three- and five-point symmetries is presented. It leads to the generic form of the symmetry generators of all the equations in this class, which satisfy a certain non-degeneracy condition. Finally… Expand
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References

SHOWING 1-10 OF 30 REFERENCES
Symmetries and first integrals of ordinary difference equations
  • P. Hydon
  • Mathematics
  • Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 2000
This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method forExpand
Factorizable Lie symmetries and the linearization of difference equations
We show that an autonomous difference equation, of arbitrary order and with one or more independent variables, can be linearized by a point transformation if and only if it admits a symmetry vectorExpand
Symmetries of Integrable Difference Equations on the Quad-Graph
This paper describes symmetries of all integrable difference equations that belong to the famous Adler Bobenko Suris classification. For each equation, the characteristics of symmetries satisfy aExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS
The study of Sophus Lie in the late nineteenth century on the unification and extension of various solution methods for ordinary differential equations led him to introduce the notion of continuousExpand
A discrete Garnier type system from symmetry reduction on the lattice
A symmetry reduction of the lattice modified Boussinesq system is studied. The full group of Lie point symmetries of the relevant system is retrieved and certain group invariant solutions areExpand
Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation
Abstract Using the direct linearization method similarity reductions of integrable lattices are constructed providing discrete analogues of the Painleve II equation. The reduction is obtained byExpand
Continuous symmetries of the lattice potential KdV equation
In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg–de Vries (lpKdV) equation. Using its associated spectral problem we construct theExpand
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding differenceExpand
Lie symmetries and the integration of difference equations
Abstract The Lie method is generalised to ordinary difference equations. We prove that the order of ordinary difference equations can be reduced by one, provided the equation under considerationExpand
...
1
2
3
...